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Contribution à l'étude des processus stochastiques sur les groupes quantiques

Abstract : In this thesis stochastique processus on Hopf algebras are studied. These algebras, also known under the name quantum group, play in important role in mathematical physics. A major part of this workis concerned with Lévy processes, i.e. processes with stationary and independent increments. Two construction are proposed, one starts from a classical Lévy process, the other one uses quantom random walks. These pro cesses are then studied with the help of dual representations and Appell systems. This has allowed us to prove an analogue of the Feynman-Kac formula, and to study the relation between the processes and their evolution equations. Dual representations are also used to give sufficient conditions for the existence of classical versions of Lévy processes on bialgebras, and to calculate the classical generators. Several examples including the Azéma martingale are treated in detail.
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Uwe Franz. Contribution à l'étude des processus stochastiques sur les groupes quantiques. Mathématiques générales [math.GM]. Université Henri Poincaré - Nancy 1, 1997. Français. ⟨NNT : 1997NAN10080⟩. ⟨tel-01748598⟩

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